Effects of interface dislocations on properties of ferroelectric thin films

ARTICLE IN PRESS

Journal of the Mechanics and Physics of Solids

55 (2007) 1661–1676

0022-5096/$ -

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Effects of interface dislocations on properties offerroelectric thin films

Yue Zhenga,b, Biao Wangb,c, C.H. Woob,�

aElectro-Optics Technology Center, Harbin Institute of Technology, Harbin 150001, ChinabDepartment of Electronic and Information Engineering, The Hong Kong Polytechnic University,

Hong Kong, SAR, ChinacThe State Key Lab of Optoelectronic Materials and Technologies, School of Physics and Engineering,

Sun Yat-sen University, Guangzhou 510275, China

Received 13 November 2006; accepted 21 January 2007

Abstract

Effects of interfacial dislocations on properties of thin-film ferroelectric materials, such as the self-

polarization distribution, Curie temperature, dielectric constant and the switching behaviors, are

investigated via the system dynamics based on the Landau–Devonshire functional. Dislocation

generation in the film is found to reduce the overall self-polarization and the Curie temperature. The

spatial variations are both very strong, particularly in the immediate neighborhood of the dislocation

cores. In agreement with previous results based on a stationary model, a dead layer exists near the

film/substrate interface, in which the average self-polarization is much reduced. Moreover, it is

evident from our results that interface dislocations play an important role in suppressing the remnant

polarization and the coercive field of the polarization.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Interface dislocations; Self polarization; Ferroelectric thin film; Remnant polarization; Coercive field

see front matter r 2007 Elsevier Ltd. All rights reserved.

.jmps.2007.01.011

nding author. Tel.: +852 2766 6646; fax: +852 2362 8439.

dresses: [email protected] (Y. Zheng), [email protected] (B. Wang),

polyu.edu.cn (C.H. Woo).

www.elsevier.com/locate/jmps

dx.doi.org/10.1016/j.jmps.2007.01.011

mailto:[email protected]



ARTICLE IN PRESSY. Zheng et al. / J. Mech. Phys. Solids 55 (2007) 1661–16761662

1. Introduction

Much work has been done recently on nano-scale ferroelectric thin films because of theirremarkable properties that make them very useful in electro-optic, pyroelectric, andpiezoelectric devices, as well as ultra-high-density nonvolatile memories (Haeni et al., 2004;Fong et al., 2004; Streiffer et al., 2002; Balzar et al., 2004). It is well known that propertiesof ferroelectrics in the bulk and thin-film forms may be significantly different, dependingon the combined effects of film surfaces, depolarization field, film/substrate interfaces,epitaxial stress and the external electric field, etc (Catalan et al., 2005; Sinnamon et al.,2002; Wesselinawa et al., 2005; Scott et al., 2005; Wang et al., 2003). In this regard, misfitdislocations introduces a large stress field superimposing on the epitaxial stresses tosignificantly affect the physical properties of the film. Related investigations constitute anactive area of thin-film research.Dislocation structure formed in BaTiO3 grown on SrTiO3 substrate has been

analyzed by using X-ray and transmission electron microscopy, and determined thecritical thickness of the misfit dislocation generation (Suzuki et al., 1999). They alsoconfirmed that the misfit relaxation depended on the presence of dislocations.Similar investigations have been carried out and result showed that misfit dislocationswere very important on the stability of the polarization field (Sun et al., 2004; Chu et al.,2004). The dielectric properties of ferroelectric thin film also depended on internalstresses and dislocation-type defects (Li et al., 2001; Canedy et al., 2000). Using thephase-field model, effects of interfacial dislocations on the polarization distributionand domain structure of ferroelectric thin film have been investigated (Hu et al.,2003). They also developed a method to predict the evolution of a domain structurein a ferroelectric thin film with an arbitrary spatial distribution of dislocations. However,the effects of the depolarization field and the near-surface eigenstrain relaxationcould not be readily taken into account within this approach. Another concern alsoarose from the non-linear nature of the model, which did not always admit a uniquesolution.In our previous work (Wang and Woo, 2005; Zheng et al., 2006a), we adopted the time-

dependent Ginzburg–Landau equation approach to investigate the self polarizationdistribution near misfit dislocations, successfully including the effects of relaxation of thesurface effect and the depolarization. The existence of a dead layer near the film/substrateinterface was confirmed (Haun et al., 1987). However, the dielectric constant was assumedto be spatially uniform. This assumption is inconsistent with the spatially varyingpolarization field, on which the dielectric constant is functionally dependent. Besides,among many of the most important properties of a ferroelectric thin film, only thespontaneous polarization distribution was discussed.In this paper, effects of interfacial dislocations on the self-polarization, Curie

temperature and dielectric constant are considered as functions of temperature and filmthickness. The spatial variations of the self-polarization, the depolarization, and theextrapolation length are specifically taken into account. This is done via the finite-difference solution of the evolution equation derived from a thermodynamic model inwhich the free energy is constructed with the help of the Landau–Devonshire functional.The interface dislocations effects on the remnant polarization and the coercive field offerroelectric thin film also is investigated. The results are discussed and the observationsconcluded.

ARTICLE IN PRESSY. Zheng et al. / J. Mech. Phys. Solids 55 (2007) 1661–1676 1663

2. Theory

2.1. The free energy equation

We consider a single-domain ferroelectric thin film on a compliant substrate, whichundergoes a cubic to tetragonal (tetragonal to cubic) phase transformation on cool-down(heat-up). We use a coordinate system in which the x-axis is parallel to [1 0 0], the y-axis to[0 1 0], and the z-axis to [0 0 1], where the film in the x- and y-directions extend to infinity,the film surface and the film/substrate interface are on the z ¼ h and z ¼ 0 planes,respectively, h being the film thickness (Fig. 1). A straight edge dislocation is placed at theorigin, with Burgers vector in the x-direction and dislocation line in the y-direction.

As in the classical description of electrical susceptibility of polar molecules, we mayidentify E, which has its origin from the combined effects of P and Eext, as the sum of theexternal field Eext and the depolarization field Ed. We call P the self-polarization instead ofthe spontaneous polarization in our previous work (Wang and Woo, 2005; Zheng et al.,2006a). The evolution of the system is then driven by the chemical potential derived from atotal free energy that can be written as the sum of self-energies due to the self-polarizationP, the induced polarization PE and the stress field. The total polarization Ptotal at any pointis considered to be the sum of a permanent molecular component P characteristic of thespecific para/ferroelectric phase it is in, and an induced component PE. The inducedcomponent PE is the polarizations (ionic+electronic) induced by the total electric field E inthe dielectric according to the law of electrostatics in the absence of free charges, r �D ¼0 ¼ r � ð�0E þ PtotalÞ ¼ r � ð�0E þ wE þ PÞ ¼ r � ðe E þ PÞ subject to the appropriateboundary conditions on the dielectric surfaces. Without external electric field (Eext ¼ 0),we have E ¼ Ed. The total polarization is sum of the self-polarization and inducedpolarization by the depolarization filed Ed, and which can be obtained by Ptotal ¼ Pþ wEd.Here w and e are the temperature-independent components of the susceptibility andpermittivity, which corresponds to the induced polarization. The same relation has alsobeen discussed for investigating size effects in expitaxial islands and films (Wang and Woo,2006; Wang and Zhang, 2006). In this work, we only establish a simple model trying torepresent the complex influence of misfit dislocations on the properties of ferroelectric thinfilms. P in ferroelectric thin film is only assumed to have a non-zero Cartesian componentonly along the z-direction, i.e., Pz ¼ P and Px ¼ Py ¼ 0.

Considering the effects of the depolarization and applied electric fields, the total freeenergy of a ferroelectric thin film with the interface dislocations, including Land-au–Devonshire energy, the gradient energy, the depolarization energy, the electrical energyinduced by external electric field. When an interface dislocation is generated in film/

Fig. 1. Schematic of the interfacial dislocation between the ferroelectric thin film and the substrate.


substrate system, the total Helmholtz free energy ~G is obtained from a Legendretransformation from a Gibbs free energy G (Pertsev et al., 1998)

~G ¼ G þ m11s11 þ m22s22 þ m33s33 þ m13s13. (2.1)

Then ~G can be written as (Wesselinawa et al., 2005; Sun et al., 2004; Li et al., 2001;Zheng et al., 2006a; Wang and Zhang, 2006),

~G ¼

ZZS

G0 þA

2ðT � Tc0ÞP

2 þB

4P4 þ

C

6P6

(þ

1

2S11ðs211 þ s222 þ s233Þ

þ S12ðs11s22 þ s11s33 þ s22s33Þ þ1

2S44ðs212 þ s213 þ s223Þ �Q11s33P2

�Q12ðs11P2 þ s22P2ÞþD44

2

qP

qx

� �2

þD11

2

qP

qz

� �2

�1

2EdP� EextP

)dxdz

þ

ZSx

D11P2

d1dxþ

ZSz

D44P2

d2dz, ð2:2Þ

where A, B, C, D11 and D44 are expansion coefficients of the Landau–Devonshirefunctional at constant stress. Tc0 is the Curie temperature of the bulk crystal. sij is the totalstress field in the film, in the presence of the interfacial dislocations, Qij is theelectrostrictive tensor (Tilley, 1996; Pertsev et al., 1998). Sz and Sx represent theupper–lower and left–right surface planes that cover the entire surface S of the film. d1 andd2 are the extrapolation lengths, which measure the surface effects on the film surface andthe film/substrate interface, respectively, including near-surface eigenstrain relaxations(Zhong et al., 1994).

2.2. The stress field induced by the interface dislocations

We use periodic boundary conditions along the x-direction to reflect a configuration inwhich the interfacial dislocations form an array lying on the z ¼ 0 plane, with CII interfacedislocation (Hu et al., 2003) and line directions parallel to the y-direction, through thecenters of identical simulation cells repeated ad infinitum along the x-direction. The surfaceeffects on Sx and Sy as well as the depolarization field along the x and y directions areneglected.The resultant stress field of the dislocation array, within isotropic linear elasticity theory,

is given by (Hirth and Lothe, 1982; Hull and Bacon, 2001)

s11ðx; zÞ ¼ �b

2pS44 1þ S12

S11

� �Xm

zð3x2

m þ z2Þ

ðx2m þ z2Þ2

,

s22ðx; zÞ ¼ �b

pS44 1þ S12

S11

� �S12

S11

Xm

z

x2m þ z2

,

s33ðx; zÞ ¼b

2pS44 1þ S12

S11

� �Xm

zðx2

m � z2Þ

ðx2m þ z2Þ2

,

s13ðx; zÞ ¼ s31ðx; zÞ


¼b

2pS44 1þ S12

S11

� �Xm

xmðx2m � z2Þ

ðx2m þ z2Þ2

,

s12ðx; zÞ ¼ s21ðx; zÞ ¼ s32ðx; zÞ ¼ s23ðx; zÞ ¼ 0, ð2:3Þ

where Sij is the elastic compliance tensor at constant polarization. Lx is the averageseparation between dislocations. b ¼ |b| is the magnitude of the Burgers vector of thedislocations, and xm ¼ x+mLx, where m ¼y�2, �1, 0, 1, 2,y, denotes the mthsimulation cell. We note that the singularity of the strain field of a dislocation in thedislocation core is caused by the break down of the linear elasticity theory in that region.Nevertheless, it is well known, through computer simulation studies that the linearelasticity strain field is a good approximation except in a small region within a few burgersvectors (�1 nm) from the dislocation center, where it is almost constant. In our calculation,we simply kept away from this small region.

2.3. The electric field in thin film

We assume that the self-polarization in the film is along the z-direction in the absence ofexternal free charge carriers. The depolarization field can be obtained by solving theelectrostatic equilibrium, X �D ¼ 0 with special boundary conditions, such as short-circuitand open-circuit boundary conditions (Li et al., 2002a; Wang and Woo, 2005; Wang andWoo, 2006). In writing Ed in terms of e, where e is the dielectric constant of the ferroelectricmaterial, we note that P is not the total polarization (Kretchmer and Binder, 1979), but theself-polarization. Nevertheless, the equivalence of the two different forms can be easilydeduced (Wang and Woo, 2005). When the ferroelectric film is sandwiched between twoshort-circuited metallic electrodes, the depolarization field Ed has been approximatelyobtained (Wang and Woo, 2005; Wang and Zhang, 2006; Lupascu, 2004; Mehta et al., 1973).

With the external field switched off, e is a function of P given by

�ðPÞ ¼q2f

qP2

� ��1, (2.4)

where f is the local energy density (Li et al., 2001). Through this relation, it can be seen that (P)and the self-polarization P at each location has to be solved self-consistently (see Section 3).

2.4. The time-dependent Ginzburg– Landau equation

From Eqs. (2.1)–(2.4), the time evolution of the self polarization in ferroelectric thin filmis governed by the time-dependent Ginzburg–Landau equation (Zheng et al., 2006a–c andLi et al., 2001),

qPðx; z;T ; tÞ

qt¼ �M

d ~GdP

¼M � A�ðx; z;T ; tÞP� BP3 � CP5

�

þD44q2Pqx2þD11

q2P

qz2� Edðx; z;T ; tÞ

�, ð2:5Þ


where M is the kinetic coefficient related to the domain wall mobility, and

A�ðx; z;T ; tÞ ¼ AðT � Tc0Þ � 2Q11s33ðx; z; tÞ

� 2Q12ðs11ðx; z; tÞ þ s22ðx; z; tÞÞ. ð2:6Þ

The electrical and mechanical boundary conditions have a significant effect onproperties of ferroelectric thin films. In this work, the electric boundary condition is(Zhong et al., 1994; Zheng et al., 2006b),

qP

qz¼ �

P

d; for z ¼ 0; h (2.7)

from the surface term in Eq. (2.2).When surface effects are negligible, qP=qz ¼ 0 at z ¼ 0 and h, corresponding to

extrapolation lengths of d0 ¼ dL!1 (Zheng et al., 2006c). The mechanical boundaryconditions for the ferroelectric thin film are taken as: the ‘‘upper’’ surface is traction-freeand the interface is determined by the interface dislocation.

3. Numerical method

3.1. The finite difference method and the Runge– Kutta method

The evolution of the self-polarization field P is obtained by numerically solving the time-dependent Ginzburg–Landau Eq. (2.5) subject to the electric and mechanical boundaryconditions. We adopt the second-order finite difference method for spatial integration andthe fourth-order Runge–Kutta method for time integration to solve Eq. (2.5) in this work(Wang and Zhang, 2006). The ferroelectric thin film is considered as a stack of layers, eachof which has a finite thickness Dz with physical properties that are assumed to be uniform.To ensure the validity of the thermodynamic description, Dz is chosen to be sufficiently largecompared with the lattice constant. N is the total number of the layers, satisfying h ¼ NDz.A layer located at a position between z and z+Dz is identified by the index j, so thatzj ¼ jDz. We also use a finite thickness Dx in the x-direction, so that any location along the x

direction can be expressed in the form xi ¼ iDx. The ferroelectric thin film is represented byrepeated simulation cells in the x-direction, each with width L ¼WDx, where W is the totalnumber of layers along the x direction. The depolarization field equation and the boundaryconditions equation can be given by the difference equations for computing effects of thedepolarization field and the near-surface eigenstrain relaxation.

3.2. The dielectric constant at each location

The local polarization P from Eq. (2.5) with the boundary conditions and thecorresponding dielectric constant e can be solved self-consistently with an iterative schemeusing a spatially homogeneous value from the experiments (Zhong et al., 1994) to calculateP as an initial estimate. Corrected values of e and P at each location in subsequentiterations then follow from � ¼ ðq2f =qP2Þ

�1, the form of which is given by

��1ðxi; zj ;TÞ ¼ A�ðxi; zj ;TÞ þ 3BP2ðxi; zj ;TÞ

þ 5CP4ðxi; zj ;TÞ. ð3:1Þ


4. Simulation results and discussions

To be specific, we consider a PbTiO3 thin film on a rigid LaAlO3 substrate. We use as anapproximation material constants for the Landau free energy, the electrostrictivecoefficients and the elastic properties for bulk materials from the literatures (Lakovlevet al., 2002; Pertsev et al., 1998; Streiffer et al., 2002; Li et al., 2002b). We consider that thisapproximation should be sufficient for our discussion.

Fig. 2. The stable polarization field in a PbTiO3 film in the presence of an interfacial edge dislocation at (0 0 0) at

different temperature of T ¼ 0–1100K.

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Fig. 3. Curie temperature around a b ¼ a½1̄ 0 0� interfacial edge dislocation at (0 0 0). Arrow indicates Curie

temperature of PbTiO3 film grown on LaAlO3 substrate without the interface dislocations.

Y. Zheng et al. / J. Mech. Phys. Solids 55 (2007) 1661–16761668

We first consider the polarization field near a b ¼ a½1̄ 0 0� single edge dislocation,neglecting the external electric field, depolarization field and near-surface eigenstrainrelaxation. The calculated self-polarization field P with an interfacial edge dislocationat (0,0,0) for a PbTiO3 film at different temperature between 0 and 1100K are shown inFig. 2. The dead zone near the dislocations in the ferroelectric state below 600K, in whichthe polarization diminishes, and the residual polarization zone near the dislocation in theparaelectric state above 900K are clearly shown. The sizes of both zones can be seen todepend on temperature. The local Curie temperature around the dislocation can beobtained by noting the locations at which the polarization vanishes. In Fig. 3, the variationof the Curie temperature near a single dislocation is shown. Although most regionsexperience a drop of the Curie temperature by several hundred degrees due to the presenceof the dislocation, it can also rise by as much as 8001 in the compressive regions and lowerby over 10001 in the tensile regions, within �2 nm from the dislocation core, giving avariation over a range of nearly 20001 in the Curie temperatures in this small region. As aresult, while the self-polarization may still persists in the compressive regions attemperatures much higher than the Curie temperature of the bulk crystal (TC0 ¼ 763K),polarization already begins to disappear in the tensile regions at temperatures much belowthe bulk TC0. Nevertheless, these results must be interpreted with caution because thelinear elastic stress field near the dislocation core in Eq. (2.3) may be much overestimated(Woo and Puls., 1977).The equilibrium thermodynamic theory of the misfit dislocations was developed by

Matthews and Blakeslee and well established by Nix (Matthews and Blakeslee, 1974; Nix,1989). The theory can only approximately predict the critical thickness of the dislocationgeneration and the dislocation density can be obtained. In this work, the critical thicknessof the interface dislocation generation is about 4 nm when ferroelectric is in paraelectricstate, the calculated dislocation separations for the 10, 20 and 50 nm-thick films are,respectively, �18, �15 and �13 nm. We must note that the interface dislocation densityand critical thickness of the interface dislocations generation should also be determined bythe polarization in ferroelectric thin film (Wang et al., 2003; Zheng et al., 2006c). In thiswork, we only assume that no additional dislocations form during ferroelectric phase

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Fig. 4. The stable polarization distribution around periodic interface at different temperature of (a)–(d) T ¼ 0,

300, 600, 900K without considering effects of the depolarization field and the surface energy. Arrows indicate the

spontaneous polarization in PbTiO3 film grown on LaAlO3 substrate without the interface dislocations.

Y. Zheng et al. / J. Mech. Phys. Solids 55 (2007) 1661–1676 1669

transition. The polarization field for different temperatures in a film with a dislocationseparation of 15 nm is calculated. Fig. 4 shows the case with the depolarization field andthe surface effect neglected (i.e., putting e ¼N ¼ d). Compared to Fig. 2, a similar deadzone in the ferroelectric state and residual polarization near the dislocation in theparaelectric state can be seen.

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Fig. 5. The stable polarization distribution around periodic interface dislocation, with effects of the

depolarization field and the surface energy taken into account, (a) d ¼ 3 nm and e ¼ constant at T ¼ 0K, (b)

d ¼ 3 nm and e(P, T) at T ¼ 0K.


In Figs. (5)–(6), we show the case when effects of the depolarization field and the surfaceeffect are taken into account. Figs. (5)–(6) show the corresponding spatial distributionof P at 0 and 300K, with these effects taken into account under various assumptions. InFigs. 5a and 6a, the calculation is performed assuming a spatially homogeneous dielectricconstant (Lakovlev et al., 2002; Zhong et al., 1994, Wang and Zhang., 2006). In Figs. 5(b)and 6(b), P and e are solved self-consistently as functions of x and z from Eqs. (2.6) and(3.1). The difference in the spatial variations of e and P, when Figs. 5(a)–(b) is comparedwith Fig. 4 at 300K, shows that both effects are to reduce the self-polarization field in thefilm. Taking into account the dependence of e on the polarization reduces its overallmagnitude at temperatures far from the Curie temperature, thus increasing the overalldepolarization field. Near the Curie temperature, the opposite happens. The presence ofthe dead layer and its location seems to be little affected by these effects.From above method, then we can obtain relation between the self-polarization and

temperature with temperature heating-up when effects of the depolarization field and theextrapolation length are taken into account. Fig. 7 shows the effect of the dislocation onthe average polarization of the film as a function of temperature under considering effectsof the depolarization. In Fig. 7(a), we compare the average self polarization at varioustemperatures with and without interface dislocations, in a PbTiO3 film epitaxially grownon a rigid LaAlO3 substrate. The presence of the dislocation can be seen to substantiallyweaken the overall polarizability of the film, particularly at higher temperatures(800KoTo1265K), where the averaged polarization drops to less than 10% of thedislocation-free case. This weakening is due to the development of the dean zone in areaswhere the Curie temperature is reduced below the ambient temperature by the dislocation

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Fig. 6. The stable polarization distribution around periodic interface dislocation, with effects of the

depolarization field and the surface energy taken into account, (a) d ¼ 3 nm and e ¼ constant at T ¼ 300K,

(b) d ¼ 3 nm and e(P, T) at T ¼ 300K.


strain field. Thus, approximately half of the thin film is occupied by dead zones at �550K.The temperature at which the average polarization drops by half of its peak value alsoreduces by 5001, from 1000 to 500K, in the presence of dislocations. In the highly strainedcore region of the dislocations, the ferroelectric phase persists at temperatures much higherthan the Curie temperature of the bulk material. Thus, when dislocations are present, thespatial distribution of the polarization is inhomogeneous and a sharp cutoff of thepolarization at the Curie temperature does not exist, unlike the case without dislocations(Fig. 7(b)). Nevertheless, since the dislocation cores only occupy a small fractional volume,the associated effects are practically negligible when averaged over the entire film. In theforgoing calculations, we note that the dislocation spacing depends on the film thickness,and the relation between the dislocation spacing and the film thickness has beensummarized (Matthews and Blakeslee, 1974; Nix, 1989). Such a comparison is also madeto consider the effect of dislocations as a function of film thickness. For the sameinterfacial dislocation line density there is a tendency that the effect is enhanced with thefilm thickness increasing due to effect of the surface effect and the depolarization field offerroelectric film (Wang and Woo, 2003; Zheng et al., 2006a, Jo et al., 2006). Nevertheless,the effect has to be considered together with the influence of film thickness and thepolarization on the interfacial dislocation density, these results have been discussed inmany works (Sharma et al., 2004; Ban and Alpay, 2003; Zheng et al., 2006b).

With short-circuit electrical boundary condition, the total polarization in ferroelectricthin film is sum of the self-polarization and induced polarization by the depolarization filedEd. The total polarization can be easily obtained by Ptotal ¼ Pþ wEd, where P and w canbe calculated by solving Eqs. (2.7–3.1) (Wang and Woo, 2005). Due to effect of the

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Fig. 7. Under considering effects of the depolarization field and the surface energy, (solid line) also considering

effect of the periodic dislocations, and (dot line) assuming film is commensurate, (a) the relation between the

average polarization and temperature. (b) The relation between the average polarization and temperature near the

phase transition of Fig. 6(a).


depolarization field, the inhomogeneous distribution P(x,z) and average polarization /PSof self polarization induced by the stress field of the interface dislocations should bemuch weakened, which also indicate that effect of the depolarization field can muchplane variation of the polarization for ferroelectric thin film with this electricalboundary condition. This behavior of the depolarization field has been discussed bytheoretical and experimental methods (Bratkovksy and Levanyuk, 2002; Chu et al.,2003). Moreover, with other electrical boundary condition, such as open-circuit (Wangand Woo, 2006) and short-circuit with incomplete compensation of the ferroelectric

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Fig. 8. Hysteresis loops of ferroelectric thin films (h ¼ 20 nm) without interface dislocations (red) and with

interface dislocations (blue).


polarization charge (Mehta et al., 1973), the distributions of the total polarization aredifferent completely due to difference of the depolarization field with these differentboundary conditions.

We compare two cases: (i) a thin film on a coherent substrate (i.e., no interfacedislocations), and (ii) a thin film with interface dislocations. With applying externalelectric field, the average polarization is Pa ¼ hPtotalðT ;EextÞi, which is determinedby the self-polarization, the depolarization and applied electric field by solving Eq. (1).Where the external electric field changes as sine function, and can be written asEext ¼ E0 sinð2pt=T 0Þ ¼ E0 sinð2ptf 0Þ, where E0, T 0 and f 0 are the amplitude, period andfrequency, respectively. Where E0 ¼ 5� 108V/m and the step time Dt ¼ 50� 10�10 s.Fig. 8 shows the corresponding hysteresis loops of PbTiO3 thin films (h ¼ 20 nm) at 300K(ferroelectric). The remnant polarization Pr and coercive field Ec all see reductions, asexpected, in the presence of dislocations because of the resulting stress relaxation andpolarization drop.

5. Summery and conclusions

The time-dependent Ginzburg–Landau equation of a ferroelectric thin film withinterfacial dislocations is solved numerically using a finite-difference scheme. Taking intoaccount the inter-dependence of the polarization and depolarization fields, as well as thesurface effect, effects of interfacial dislocations on ferroelectric thin films are investigated.We obtain the self-polarization field around a single edge misfit dislocation at differenttemperatures, and determine the dead region (i.e. where P ¼ 0) for various temperatures.The results allow us to map out the Curie temperature at every location in the film aroundthe dislocation as a function of x and z. In this calculation, the dielectric constant e iscalculated self-consistently with the self-polarization P as a function of T, x and z, via thefree energy of the system. The value of the dielectric constant is obtained numerically viaan iterative procedure.


Our results show that misfit dislocations have very important effects on the self-polarization field and the local dielectric properties in thin-film ferroelectrics, particularlynear the dislocations. The existence of a dead layer near the film/substrate interface isconfirmed, where self-polarization is much reduced. The extent of the dead layer is foundto be temperature dependent. As a result, the presence of interfacial dislocationssignificantly reduces the average polarization of ferroelectric thin film at hightemperatures, which causes the lowering of the effective Curie temperature by hundredsof degrees. This significant reduction of self-polarization by the presence of misfitdislocations at higher temperatures is seen in all film thicknesses up to 100 nm. Beyond theCurie temperature, however, self-polarization does not vanish completely in the presenceof misfit dislocations. In the present calculation, the effects of the depolarization fields aswell as the near-surface eigenstrain relaxation are both found to be relatively unimportant.Our results also show that the presence of interfacial dislocations significantly affects thespatial distribution of the polarization and reduces remnant polarization and thecoercive field because of stress relaxation. Even at zero average polarization, the spatialheterogeneity of the polarization can be very large. In future works, we will consider thatthe presence of inhomogeneous elastic fields created by the misfit dislocations, theself-polarization and total polarization distribution in the film inevitably becomes non-uniform too. In this situation, the in-plane polarization component P1 and P2 mayappear in the film (owing to the depolarization field and electrostriction) even when thehomogeneously strained film has only the out-of-plane component P3. At the sametime, the dislocation density and the Burgers vector will change (increase and turn ormove) due to coupling relation between the polarization and the stress field induced by theinterface dislocation. Properties, such as phase diagram, domain patterns and tenabilityetc., will be investigated under consider general these cases.

Acknowledgments

This project was supported by Grants PolyU5312/03E, 5322/04E and GU164. Co-author BW is also grateful for grants from the National Science Foundation of China(Nos. 50232030, 10172030 and 10572155) and the Science Foundation of GuangzhouProvince (2005A10602002).

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Effects of interface dislocations on properties of ferroelectric thin films